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Proof step 11 from proof modmod-add-if3 (previous step)

i0 < 0, 0 < j, i + i0 < 0, i + i0 % j < 0, 0 ≤ i ⊦ (i + i0) % j = (0 ≤ i + i0 % j ∧ (i + i0 % j) % j ≠ 0 ⊃ (i + i0) % j + j;(i + i0) % j)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: modmod-add2-posneg-neg: ⊦ i + i0 % j < 0 ∧ 0 ≤ i ∧ i0 < 0 ∧ 0 < j → (i + i0 % j) % j = (i + i0) % j;% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i + j < j ↔ i < 0; (s)%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2